Let be the semilattice of degrees of recursive unsolvability. The main result of this paper is that the first-order theory of is recursively isomorphic to the truth set of second-order arithmetic (Corollary 5.6). We also obtain a strong result concerning first-order definability in where j is the jump operator (Theorem 3.12). The structure of has been investigated strenuously by Kleene and Post [12], Spector [29], Sacks [20], Lerman [15] and a host of others. The first-order theory of has been commented upon from time to time by various authors including Jockusch and Soare [10], Miller and Martin [17], Rogers [19], Shoenfield [24], [26] and Stillwell [30]. Our main result can be regarded as a refinement of the theorem of Lachlan [13] that the first-order theory of is undecidable. Like Lachlan we use initial segments, but we combine them with the jump operator (Theorem 2.1). Our curiosity about the subject of this paper was first awakened in 1969 by Gerald E. Sacks who asked whether the first-order theory of is hyperarithmetical (see also Problem 70 in [5]). We are also grateful to Carl G. Jockusch, Jr. for timely expressions of interest in this work. We use c to denote the set of nonnegative integers {0, 1, 2, ... }. Letters such as i, j, k, m, n denote elements of co. We write 20 for the set of totally defined, {0, 1}-valued functions on w. Letters such as f, g, h denote elements of 20. We write f 0 g for the unique function h such that h(2n) = f(n) and h(2n + 1) = g(n) for all n e w. The jump of f e 20 is f* which is again an element of 2w. Finite iterates of * are defined by f(') = f and f(nPl) = (f (n))*